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MODALITY, SI! MODAL LOGIC, NO!

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

1997 Mar 18, 5:23 p.m.

Abstract

This article is oriented toward the use of modality in artiﬁcial

intelligence (AI). An agent must reason about what it or other agents

know, believe, want, intend or owe. Referentially opaque modalities

are needed and must be formalized correctly. Unfortunately, modal

logics seem too limited for many important purposes. This article

contains examples of uses of modality for which modal logic seems

inadequate.

I have no proof that modal logic is inadequate, so I hope modal

logicians will take the examples as challenges.

Maybe this article will also have philosophical and mathematical

logical interest.

Here are the main considerations.

Many modalities: Natural language often uses several modalities in a sin-

gle sentence, “I want him to believe that I know he has lied.” [Gab96]

introduces a formalism for combining modalities, but I don’t know

whether it can handle the examples mentioned in this article.

New Modalities: Human practice sometimes introduces new modalities on

an ad hoc basis. The institution of owing money or the obligations the

Bill of Rights imposes on the U.S. Government are not matters of basic

Introducing new modalities should involve no more fuss than

logic.

introducing a new predicate.

In particular, human-level AI requires

that programs be able to introduce modalities when this is appropriate,

e.g. have function taking modalities as values.

Knowing what: “Pat knows Mike’s telephone number” is a simple exam-

ple. In [McC79b], this is formalized as

knows(pat, T elephone(M ike)),

where pat stands for the person Pat, M ike stands for a standard con-

cept of the person Mike and T elephone takes a concept of a person into

a concept of his telephone number. We might have

`telephone(mike)`

= `telephone(mary)`

,

expressing the fact that Mike and Mary have the same telephone, but

we won’t have

T elephone(M ike) = T elephone(M ary),

which would assert that the concept of Mike’s telephone number is the

same as that of Mary’s telephone number. This permits us to have

¬knows(pat, T elephone(M ary)).

even though Pat knows Mike’s telephone number which happens to be

the same as Mary’s. The theory in [McC79b] also includes functions

from some kinds of things, e.g. numbers or people, to standard concepts

of them. This permits saying that Kepler did not know that the number

of planets is composite while saying that Kepler knew that the number

we know to be the number of planets (9) is composite.

The point of this example is not mainly to advertise [McC79b] but to

advocate that a theory of knowledge must treat knowing what as well

as knowing that and to illustrate some of the capabilities needed for

adequately using knowing what.

Presumably

knows(pat, T elephone(M ike))

could be avoided by writing

(∃x)(knows(pat, T elephone(M ike) = x)),

but the required “quantifying in” is likely to be a nuisance.

Proving Non-knowledge [McC78] formalizes two puzzles whose solution

requires inferring non-knowledge from previously asserted non-knowledgeand from limiting what is learned when a person hears some information.1[McC78] uses a variant of the Kripke accessibility relation, but here

it is used directly in ﬁrst order logic rather than to give semantics to

a modal logic. The relation is A(w1, w2, person, time) interpreted as

asserting that in world w1, it is possible for person that the world is

w2. Non-knowledge of a term in w1 is e.g. the color of a spot or the

value of a numerical variable, is expressed by saying that there is a

world w2 in which the value of the term diﬀers from its value in w1.

[Lev90] uses a modality whose interpretation is “all I know is . . ..”. He

uses autoepistemic logic [Moo85], a nonmonotonic modal logic. This

seems inadequate in general, because we need to be able to express “All

1The three wise men puzzle is as follows:

A certain king wishes to test his three wise men. He arranges them in a circle so that

they can see and hear each other and tells them that he will put a white or black spot on

each of their foreheads but that at least one spot will be white. In fact all three spots are

white. He then repeatedly asks them, “Do you know the color of your spot?” What do they

answer?

The solution is that they answer, “No,” the ﬁrst two times the question is asked and

This is a variant form of the puzzle which avoids having wise men reason about how

answer “Yes” thereafter.

fast their colleagues reason.

Here is the Mr. S and Mr. P puzzle:

Two numbers m and n are chosen such that 2 ≤ m ≤ n ≤ 99. Mr. S is told their

sum and Mr. P is told their product. The following dialogue ensues: Mr. P: I don’t

know the numbers.

Mr. S: I knew you didn’t know. I don’t know either.

Mr. P: Now I know the numbers.

Mr S: Now I know them too.

In view of the above dialogue, what are the numbers?

I know about the value of x is . . ..” 2 Here’s an example. At one stage

in Mr. S and Mr. P, we can say that all Mr. P knows about the value

of the pair is their product and the fact that their sum is not the sum

of two primes.

[KPH91] treats the question of showing how President Bush could rea-

son that he didn’t know whether Gorbachev was standing or sitting and

how Bush could also reason that Gorbachev didn’t know whether Bush

was standing or sitting. The treatment does not use modal logic but

rather a variant of circumscription called autocircumscription proposed

by Perlis [Per88].

Joint knowledge and learning In the wise men problem, they learn at

each stage that the others don’t know the colors of their spots, and in

Mr. S and Mr. P they learn what the others have said. In each case the

learning is joint knowledge, wherein several people knowing something

jointly implies not only that each knows it but also that they know it

jointly. [McC78] treats joint knowledge by introducing pseudo-persons

for each subset of the real knowers. The pseudo-person knows what

the subset knows jointly. The logical treatment of joint knowledge in

[McC78] makes the joint knowers S5 in their knowledge. I don’t know

whether a more subtle axiomatization would avoid this.

[McC78] treats learning a fact by using the time argument of the ac-

cessibility relation. After person learns a fact p the worlds that are

possible for him are those worlds that were previously possible for him

and in which p holds. Learning the value of a term is treated similarly.

Other modalities [McC79a] treats believing and intending and [McC96]

treats introspection by robots. Neither paper introduces enough for-

malism to provide a direct challenge to modal logic, but it seems to me

that the problems are even harder than those previously treated.

Acknowledgements: This work was supported in part by DARPA (ONR)

grant N00014-94-1-0775. Tom Costello provided some useful discussion.

2Halpern and Lakemeyer in [HL95] show that the quantiﬁed version of Levesque’s logic

is incomplete, but this is a diﬀerent complaint from the one we make here.

References

[Gab96] Dov Gabbay. Fibred semantics and the weaving of logics: Part

I: Modal and intuitionistic logics. Journal of Symbolic Logic,

61(4):1057–1120, 1996.

[HL95]

Joseph Y. Halpern and Gerhard Lakemeyer. Levesque’s axiom-

atization of only knowing is incomplete. Artiﬁcial Intelligence,

74(2):381–387, 1995.

[KPH91] Sarit Kraus, Donald Perlis, and John Horty. Reasoning about

ignorance: A note on the Bush-Gorbachev problem. Fundamenta

Informatica, XV:325–332, 1991.

[Lev90]

Hector J. Levesque. All I know: a study in autoepistemic logic.

Artiﬁcial Intelligence, 42:263–309, 1990.

[McC78]

John McCarthy.

knowledge3, 1978. Reprinted in [McC90].

Formalization of

two puzzles

involving

[McC79a] John McCarthy. Ascribing mental qualities to machines4. In Mar-

tin Ringle, editor, Philosophical Perspectives in Artiﬁcial Intelli-

gence. Harvester Press, 1979. Reprinted in [McC90].

[McC79b] John McCarthy. First Order Theories of Individual Concepts and

Propositions5. In Donald Michie, editor, Machine Intelligence, vol-

ume 9. Edinburgh University Press, Edinburgh, 1979. Reprinted

in [McC90].

[McC90]

John McCarthy. Formalization of common sense, papers by John

McCarthy edited by V. Lifschitz. Ablex, 1990.

[McC96]

John McCarthy. Making Robots Conscious of their Mental

States6. In Stephen Muggleton, editor, Machine Intelligence 15.

Oxford University Press, 1996.

3http://www-formal.stanford.edu/jmc/puzzles.html

4http://www-formal.stanford.edu/jmc/ascribing.html

5http://www-formal.stanford.edu/jmc/concepts.html

6http://www-formal.stanford.edu/jmc/consciousness.html

[Moo85] Robert C. Moore. Semantical considerations on nonmonotonic

logic. Artiﬁcial Intelligence, 25(1):75–94, January 1985.

[Per88]

Donald Perlis. Autocircumscription. Artiﬁcial Intelligence,

36:223–236, 1988.

/@steam.stanford.edu:/u/jmc/w97/modality1.tex: begun Sat Mar 1 11:21:20 1997, latexed March 18, 1997 at 5:23 p.m.